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Properties

Label 2-210-105.53-c1-0-14
Degree $2$
Conductor $210$
Sign $-0.544 + 0.838i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)2-s + (0.680 − 1.59i)3-s + (−0.866 − 0.499i)4-s + (1.30 − 1.81i)5-s + (−1.36 − 1.06i)6-s + (−2.54 + 0.736i)7-s + (−0.707 + 0.707i)8-s + (−2.07 − 2.16i)9-s + (−1.41 − 1.73i)10-s + (1.83 + 1.05i)11-s + (−1.38 + 1.03i)12-s + (3.28 + 3.28i)13-s + (0.0535 + 2.64i)14-s + (−2.00 − 3.31i)15-s + (0.500 + 0.866i)16-s + (−1.40 + 0.375i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.392 − 0.919i)3-s + (−0.433 − 0.249i)4-s + (0.583 − 0.812i)5-s + (−0.556 − 0.436i)6-s + (−0.960 + 0.278i)7-s + (−0.249 + 0.249i)8-s + (−0.691 − 0.722i)9-s + (−0.447 − 0.547i)10-s + (0.553 + 0.319i)11-s + (−0.399 + 0.300i)12-s + (0.910 + 0.910i)13-s + (0.0143 + 0.706i)14-s + (−0.517 − 0.855i)15-s + (0.125 + 0.216i)16-s + (−0.339 + 0.0910i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.544 + 0.838i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.544 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678388 - 1.24939i\)
\(L(\frac12)\) \(\approx\) \(0.678388 - 1.24939i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (-0.680 + 1.59i)T \)
5 \( 1 + (-1.30 + 1.81i)T \)
7 \( 1 + (2.54 - 0.736i)T \)
good11 \( 1 + (-1.83 - 1.05i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.28 - 3.28i)T + 13iT^{2} \)
17 \( 1 + (1.40 - 0.375i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.72 + 2.72i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.50 + 1.47i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 7.20T + 29T^{2} \)
31 \( 1 + (0.528 - 0.914i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.22 - 1.66i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.86iT - 41T^{2} \)
43 \( 1 + (-1.79 - 1.79i)T + 43iT^{2} \)
47 \( 1 + (2.87 - 10.7i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.73 + 6.47i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.63 - 4.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.11 - 7.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.43 + 9.10i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 + (1.20 - 0.323i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (13.8 - 7.99i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.47 - 9.47i)T - 83iT^{2} \)
89 \( 1 + (-0.199 - 0.346i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.63 - 8.63i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.21016447124037630769287774452, −11.44829658658045874653451707703, −9.800895672142326572613507697377, −9.200152893184316433875254651589, −8.335013901418616631309255888469, −6.68475198919059474287025071019, −5.92234909971056374541356366078, −4.25955491416647513176334385804, −2.73999645696210048333910593634, −1.31648073778486934019502718281, 3.05389201278502411447998620170, 3.89514048619302731619017324415, 5.62520986265980991961831395881, 6.32973489281178667347769526696, 7.65571552067650673931094671198, 8.816672153479102429919897637845, 9.842364778970418673284290426298, 10.38103840819715642732026923246, 11.61539012154170066187532833431, 13.13552035410422884800518133858

Graph of the $Z$-function along the critical line